## Is mathematics discovered or invented

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• 189
where is this "Platonic world", where the plans for the Universe are stored until they are needed? For if the map/plan exists, and this Platonic world is where it exists, then where is this Platonic world? Your surmise seems to rest upon your having an answer to this question, doesn't it? :wink: :chin:

Of course I have! I was only waiting for somebody to ask me to announce the truth to the world :smile: :joke:

The answer of course is based on modern weird physic theories that nobody really understands or is able to verify (string theory, multiverse, 11-dimensional space-time, and similars), so you can use them explain whatever you want!

So, here we go:

- First of all: space-time is "emergent" (https://www.preposterousuniverse.com/quantumspacetime/index.php?title=Emergent_space(time)). This means that what we perceive as space-time could be an "object" (still physical) that has a different metric and even a different number of dimensions (holographic principle: https://en.wikipedia.org/wiki/Holographic_principle). So, there is no way decide which one of the two realities (the ologram the image that it represents) is the really "real" one. It's something similar to what happens with electromagnetic fields in special relativity: the same "thing" can be seen as an electric field by an observer and as a magnetic field by another observer.

- Second: the universe (space-time, elementary particles, forces) that we observe today are the result of an evolution from a much more symmetric (simple) "object", and the forms of what we see today are the result of "choices" of some forms instead of others (https://en.wikipedia.org/wiki/Spontaneous_symmetry_breaking). This is similar to what happened with the evolution of life: animals developed into more and more complex forms starting from much simpler inanimate matter.

So, here's the theory: there is exists this very simple "object" that is still there, at the beginning of time (but we don't see it, because from our point of view we can see only contemporary time) and a map, or plan (the "plan" :joke:) made of all possible mathematical objects that can be built. And for some strange reason that nobody knows, from the simple object "emerge" all the forms contained in the map in different quantities: the more symmetrical of them (respecting patterns) are the most common. Obviously, we can see now only part of them, but the global form of the map is some kind of fractal (like giant Mandelbrot set) in which the most common structures are present in all places and at every scale. And of course (to answer the where it exists question) the "emergent" objects are not objects of the physical 3-dimensional space, but what we see as space is only one of them.

P.S. Please don't ask me what's the meaning of the worlds in quotation marks: I have no idea!
• 8k
To believe that maths was discovered is to mistake the map for the territory. [ It also avoids the rather obvious criticism that there is nowhere in the real universe that we can point to and say "there's maths" or "Oh, look, there's a sine wave". ]

The problem with this criticism is that mistakenly equates what is 'out there', in other words what exists in the manifold domain of objects, with the real universe.

But think about this point: that in order to map and understand what is 'out there' - creating the map, that you say is 'mistaken for the territory' - maths itself has proven indispensable. And maths, furthermore, enables science to predict and discover things which could never be known by observation alone.

There are many such cases in modern mathematical physics, such as Dirac's discovery of anti-matter. 'Dirac realised that his relativistic version of the Schrödinger wave equation for electrons predicted the possibility of anti-electrons.' He said, at the time he wrote the paper, 'they must be there', and lo and behold, they were demonstrated before too long.

Eugene Wigner, who has already been cited as author of the celebrated paper in mathematical philosophy, The Unreasonable Effectiveness of Mathematics in the Natural Sciences, won the Nobel Prize 'through the discovery and application of fundamental symmetry principles in sub-atomic physics'. This is what lead him to write that paper (in which, I note, the word 'miracle' occurs 12 times.)

So I think it's not feasible to argue that the relationship between mathematics and nature is merely fortuitous. There seems a deep connection. I think the conceptual problem arises as a consequence of the tendency to regard what is real, as solely what exists in time and space. Hence the question, 'where is this so-called Platonic realm'? And the answer is that it's not in any place; but 'the domain of natural numbers' is nevertheless real, because some numbers are included in it, and others (i.e. the square root of -1) are not.

So there you have an argument for something that is real, but that doesn't exist 'out there anywhere' - which is precisely what makes it difficult to wrap your head around. Our culture is profoundly alienated from such conception (although some mathematical physicists, like Kurt Godel, Roger Penrose and Max Tegmark among others, tend towards platonism.)

There's an interesting SEP article on Platonism in the Philosophy of Mathematics which notes that:

Mathematical platonism has considerable philosophical significance. If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For platonism entails that reality extends far beyond the physical world and includes objects which aren’t part of the causal and spatiotemporal order studied by the physical sciences. Mathematical platonism, if true, will also put great pressure on many naturalistic theories of knowledge. For there is little doubt that we possess mathematical knowledge. The truth of mathematical platonism would therefore establish that we have knowledge of abstract (and thus causally inefficacious) objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.

And in fact this anomaly has been noticed by analytic philosophy, which has felt obliged to mount a defense of the 'uncanny efficacy of mathematics' whilst still trying to maintain that it is something that can be understood naturalistically (see this article):

Standard readings of mathematical claims entail the existence of mathematical objects. But, our best epistemic theories seem to debar any knowledge of mathematical objects.

And what our 'our best epistemic theories'? Why, those supported by neo-darwinian materialism! Hence the argument! But they're never going to resolve the dilemma, in my view, on the grounds that number is real, but transcendent in respect to the physical. So maths isn't explained by naturalism, as it transcends naturalism - but as we've basically debarred 'the transcendent' from modern philosophical discourse, then we can't accommodate the idea.
• 1.6k
there is exists this very simple "object" [...] and a map, or plan [...] made of all possible mathematical objects that can be built.

And where does this map exist? Where is the 'place' where this map is stored and retained, ready for later use? The only thing I know of that can store an idea is a conscious mind. Perhaps there is some other container that can also achieve this, but what and where is it, this store?
• 1.6k
in order to map and understand what is 'out there' - creating the map, that you say is 'mistaken for the territory' - maths itself has proven indispensable. And maths, furthermore, enables science to predict and discover things which could never be known by observation alone.

Maths is indispensable because it's a good and well-crafted tool. It's useful. Also in its predictive power, as you say. For those without satnavs, (real physical) maps are equally indispensable, except for those that are navigating a landscape they know pretty well. Maps are good and useful things, and there's no shame in being the map, not the territory. But a motorway is not a blue line on a paper base. A motorway is a real road, and the blue line on the map is a partial representation of a motorway, drawn for the purpose of aiding navigation. And so it does. The map is a valuable thing. It does not contain actual motorways, but suffers no loss of value as a result.

So I think it's not feasible to argue that the relationship between mathematics and nature is merely fortuitous.

No, not fortuitous. We created maths to be what it is, to do what it does. We made this valuable mapping tool on purpose. It wasn't just luck or random chance.
• 1.6k
number is real, but transcendent in respect to the physical

I do not argue against transcendence, but I wonder where it exists, outside of our minds? And I wonder if my answer is "nowhere"?
• 189
And where does this map exist? Where is the 'place' where this map is stored and retained, ready for later use? The only thing I know of that can store an idea is a conscious mind. Perhaps there is some other container that can also achieve this, but what and where is it, this store?

An idea is arbitrary data, such as a picture. For this you need a memory to store information, because there are a lot of possible combinations of shapes and colors (or pixels, if you want) that can form a picture, and you don't know which one of them is the one contained in the memory. But often interesting mathematical objects need much less information to be stored that it would seem: the Mandelbrot set is a very good example: just a couple of formulas encode an apparently infinite quantity of shapes. In a sense, the Mandelbrot formula is an excellent image-compression algorithm.
Now the question is: how much information is needed to store all possible interesting mathematical objects? If you want to describe one of them, you need information about which one is it. But to describe all of them, from my point of view this is like describing all possible books that can be written in every possible language: it's just "the combination of all strings of limited length", and this is a complete description.
Now, my idea is: if "interesting" mathematical objects are in some way identifiable by a simple concept, it's conceivable that the description of "all of them" doesn't need information to at all to be described.
I don't believe that the physical universe is "made" of mathematical objects, but that laws of physic in some way favor the development of objects that "resemble" the mathematical structures that we judge "interesting". Why that coincidence? Because the same laws of physics are at the base of the evolution of human mind: in some way the human mind "favours" the recognition of the same structures, because these structures are "favoured" in the whole universe, and our mind is part of it.
Of course there is an enormous quantity of "information" in the physical world around us, and the fact that it's map contains no information doesn't seem to make sense. But the information that we see could be rather related to our very particular "position" inside the universe (something like living in a very narrow subset of the Mandelbrot set).
However this is only a vague idea: you should define how to measure this "information", and even in that case, this doesn't explain where the information about the laws of physics is contained, and probably a thousand of other things that make no sense.
• 8k
Maths is indispensable because it's a good and well-crafted tool. It's useful.

It's not just a tool, or rather, if it is, it's a meta-tool, something used to make tools. Knowing maths, you can make tools you could otherwise not:

Without it, you might have to content yourself with something like this:

We created maths to be what it is, to do what it does. We made this valuable mapping tool on purpose

You haven't really advanced an argument for that, though. I think I answered your initial assertion of this point, in terms of the argument that through maths, we can discover many real principles and properties, on the basis of which you can then invent all kinds of devices - like the LHC above. But the things discovered, like natural laws, are plainly not invented by us, and their mathematical qualities are likewise there to be found.

But it's too cheap to just say 'we make it up' - we continually have to validate mathematical ideas, not only against other mathematical ideas, but also against real-world predictions. And so I say that in some fundamental sense, this amounts to a process of discovery, not simply invention, although there are elements of both.

Regarding the transcendent nature of number: number is transcendent in the Kantian sense that it is essential to the operation of reason, whilst not necessarily being explicable by reason. There is no consensus on the nature of number; it is far from easy to understand or explain what it is, precisely, and there are many theories or schools of thought. Yet we use number, often instinctively, to understand and explain many other things.
• 1.6k
I think I answered your initial assertion of this point, in terms of the argument that through maths, we can discover many real principles and properties, on the basis of which you can then invent all kinds of devices - like the LHC above. But the things discovered, like natural laws, are plainly not invented by us, and their mathematical qualities are likewise there to be found.

Yes, we can "discover many real principles and properties" using maths to help, but that does not show maths to be discovered, only the "principles and properties" you refer to.
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